Generate Particles with Random colors (continuously or from a set)
I'm going to keep this short and simple. Do everything Duane Dibbley did but use an Object Info node instead of a Particle Info node. Connect the random output directly to the color ramp, no math node needed. 



Pro Tip: Make sure that if you are using a color ramp set to constant that you don't put any colors at position 1 (all the way to the right). They won't end up being used.